1. Field of the Invention
The present invention relates to a constant-speed scanning lens system to be used in a flat-surface scanning optical system.
2. Description of the Prior Art
As a flat-surface scanning optical system, an optical system as shown in FIG. 1 is used. In this optical system, the light beam from the light source 1 such as a laser means is reflected by a polygonal rotating mirror 2 (an oscillating mirror is sometimes used instead) and, then, imaged on a scanning surface 4 by means of an imaging lens 3. When the polygonal rotating mirror 2 is rotated in the direction of arrowhead A, the light beam spot moves in the direction of arrowhead B on the scanning surface 4. At that time, the light beam spot should move at a constant speed on the scanning surface. Therefore, as the imaging lens for a flat-surface scanning optical system, it is impossible to use a general photographic lens or the like. In case of such lenses as general photographic lenses for which distortion is corrected favourably, the relation among the field angle .theta., focal length f and image height y is expressed by the following formula (i). EQU y=f.multidot.tan .theta. (i)
In other words, the image height y (the distance from the center of scanning surface to the beam spot in case of beam-spot scanning) varies in proportion to tan .theta.. When, therefore, the polygonal rotating mirror is rotated at a constant speed, the moving speed of beam spot becomes higher when the beam spot comes nearer the margin of scanning surface because the varying rate of .theta. is constant. As a countermeasure for eliminating the above disadvantage so that the beam spot moves at a constant speed on the scanning surface, it may be theoretically possible to adequately control the rotational speed of polygonal rotating mirror instead of rotating it at a constant speed. However, this method is not practicable because there are many difficult problems in practice.
Therefore, in the known method, it is arranged to move the beam spot at a constant speed by using a lens called an f.theta. lens which is different from general photographic lenses. The f.theta. lens is arranged so that the afore-mentioned image height y, focal length f and field angle .theta. have the relation expressed by the following formula (ii). EQU y=f.multidot..theta. (ii)
When a lens which satisfies the above formula (ii) is used, y varies at a constant rate when the polygonal rotating mirror rotates at a constant speed and, therefore, the beam spot moves at a constant speed.
The above formula (ii) is satisfied when the lens is designed based on general photographic lenses or the like so that barrel distortion is purposely caused.
Generally, distortion D is expressed by the following formula. ##EQU1##
In the above formula, reference symbol Y' represents the actual image height. General photographic lenses and the like are usually designed so that D becomes a value near zero.
On the other hand, in case of an f.theta. lens, aberration is corrected so that D' expressed by the following formula becomes zero. ##EQU2##
For an actual f.theta. lens, it is desired that the beam diameter is small and, consequently, the beam spot diameter is small in addition to the fact that the scanning speed should be constant, i.e., the afore-mentioned formula (ii) should be satisfied. The beam spot diameter d is expressed by the following formula (iii) where reference symbol F represents the F number of the lens system, reference symbol .lambda. represents the wavelength of the light to be used, and reference symbol k represents the constant. EQU d=k.multidot.F.multidot..lambda. (iii)
In the above formula (iii), k and .lambda. are pre-determined values and, therefore, only F can be varied when designing the lens system. To make the beam spot diameter small, it is therefore necessary to make F small.
On the other hand, to correct offaxial aberrations favourably, it is preferable that .theta. is smaller. To make .theta. small when the size of scanning surface is pre-determined, it is necessary to make f large. However, to obtain the lens system of the same F number, it is necessary to make the lens diameter large when f is made large.
When .theta. is made small, it is necessary to make the number of faces n of polygonal rotating mirror large. Besides, when f is made large, it is necessary to make the size of each face of polygonal rotating mirror large. This is due to the following reason. That is, as the F number should be small as explained in the above, it is necessary to make the lens diameter large when f is made large and, consequently, the beam diameter becomes large. Therefore, it becomes necessary to make the size of each face of polygonal rotating mirror large. As the size of polygonal rotating mirror is decided by the number of faces and size of each face, the polygonal rotating mirror should be made doubly large in order to make .theta. small and f large as described in the above. When the polygonal rotating mirror is large, it is difficult to manufacture it accurately and its cost becomes high. Therefore, it is not desirable to make the polygonal rotating mirror large.
The relation between .theta. and number of faces n of polygonal rotating mirror is expressed by the following formula (iv). EQU .theta.&lt;360.degree./n (iv)
The effective scanning rate in a scanning system is expressed by the following formula. ##EQU3## This corresponds to the ratio between the length of scanning surface and scanning length by each face of polygonal rotating mirror. It is not preferable to make the scanning length by each face of polygonal rotating mirror unnecessarily long. Therefore, it is not preferable to make the value of n smaller than n.sub.0, which represents the maximum value of n when .theta. in the formula (iv) is a certain value, because the efficiency becomes low. Therefore, to make the value of n small, it is necessary to make the value of .theta. as large as possible, and a wide-angle lens is required as the f.theta. lens.
Due to the above-mentioned reasons, it is preferable that the f.theta. lens has a small F number, wide field angle, compact size and favourably corrected offaxial aberrations.
Out of known f.theta. lenses, there are telecentric lens systems. For those f.theta. lenses, however, the lens diameter larger than the size of image surface is required and, therefore, the lens becomes extremely large.
Besides, f.theta. lenses which are not telecentric lens systems are also known. However, those f.theta. lenses have extremely large F numbers, i.e., about F/60, and it is impossible to obtain a small beam spot.